#2773728

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,4)(3,7,5,8)(10,12)(11,15)(27,31)(29,32);(1,3,6,5)(2,7,4,8)(9,10,14,12)(11,13,15,16)(17,18,21,20)(19,23,24,22)(25,26,28,30)(27,29,31,32);(1,3,6,5)(2,8,4,7)(9,11,14,15)(10,13,12,16)(17,19,21,24)(18,22,20,23);(1,4,6,2)(3,7,5,8)(9,12,14,10)(11,13,15,16)(25,27)(26,29)(28,31)(30,32);(1,5,6,3)(2,7,4,8)(9,13)(10,15)(11,12)(14,16)(17,20,21,18)(19,23,24,22)

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=gh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2680924);